University of Misan College of Engineering CONTROL | Electrical Engineering Department [|á{tÅ TÄàt| 45 Chapter Four Time‐Domain Analysis of control system The time response of a control system consists of two parts: the transient response and the steady-state response. By transient response, we mean that which goes from the initial state to the final state. By steady-state response, we mean the manner in which the system output behaves as (t) approaches infinity. Thus the system response e(t) may be written as: ܥሺݐሻൌ ܥ௧ ሺݐሻ ܥ௦௦ Where: ܥ௧ ሺݐሻ……… ݎݐݏݎݐݏݏ. ܥ௦௦ ሺݐሻ……… ݐݏݎݕݏݏ. Absolute Stability, Relative Stability, and Steady-State Error. In designing a control system, we must be able to predict the dynamic behavior of the system from a knowledge of the components. The most important characteristic of the dynamic behavior of a control system is absolute stability—that is, whether the system is stable or unstable. A linear control system is stable if the output eventually comes back to its equilibrium state when the system is subjected to an initial condition. A linear control system is critically stable if oscillations of the output continue forever. It is unstable if the output diverges without bound from its equilibrium state when the system is subjected to an initial condition. The transient response of a practical control system often exhibits damped oscillations before reaching a steady state. If the output of a system at steady state does not exactly agree with the input, the system is said to have steady state error.
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University of Misan
College of Engineering CONTROL | Electrical Engineering
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Chapter Four
Time‐Domain Analysis of control system
The time response of a control system consists of two parts: the transient response
and the steady-state response. By transient response, we mean that which goes from
the initial state to the final state. By steady-state response, we mean the manner in
which the system output behaves as (t) approaches infinity. Thus the system
response e(t) may be written as:
Where:
……… .
……… .
Absolute Stability, Relative Stability, and Steady-State Error.
In designing a control system, we must be able to predict the dynamic behavior of
the system from a knowledge of the components. The most important characteristic
of the dynamic behavior of a control system is absolute stability—that is, whether
the system is stable or unstable. A linear control system is stable if the output
eventually comes back to its equilibrium state when the system is subjected to an
initial condition. A linear control system is critically stable if oscillations of the
output continue forever. It is unstable if the output diverges without bound from its
equilibrium state when the system is subjected to an initial condition.
The transient response of a practical control system often exhibits damped
oscillations before reaching a steady state. If the output of a system at steady state
does not exactly agree with the input, the system is said to have steady state error.
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This error is indicative of the accuracy of the system. In analyzing a control system,
we must examine transient‐response behavior and steady‐state behavior.
The typical test signals for time response of control system are:
Typical Test Signals. The commonly used test input signals are step functions,
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4‐ Parabolic input:
Transient response:
1- FIRST-ORDER SYSTEMS
Physically, this system may represent an RC circuit, thermal system, or the
like A simplified block diagram is shown in Figure below.
The input-output relationship is given by
R(t)=t2
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We will analyze the system responses to such inputs as the unit-step, unit-ramp, and unit-impulse functions. The initial conditions are assumed to be zero. a‐ Unit-Step Response of First-Order Systems. Since the Laplace
transform of the unit-step function is 1/s, substituting to the
above equation.
Expanding into partial fractions gives.
Taking the inverse Laplace transform
Above equation states that
initially the output is zero and finally it becomes unity.
exponential response curve at the value of is 0.632,
or the response has reached 63.2% of its total change.
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the exponential response curve is that the slope of the tangent line at
0 1/ ,
The smaller the time constant T, the faster the system
response.
b‐ Unit-Ramp Response of First-Order Systems. Since the
Laplace transform of the unit-ramp function is 1/ 2
Expanding into partial fractions gives
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Taking the inverse Laplace transform
The error signal is then
The error in following the unit-ramp input is equal to T for sufficiently
large .The smaller the time constant T, the smaller the steady-state
error in following the ramp input.
c‐ Unit-Impulse Response of First-Order Systems. For the unit-
impulse input, 1
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The inverse Laplace transform
2- SECOND-ORDER SYSTEMS
Here we consider a servo system as an example of a second-order system.
Servo System. The servo system shown in Figure below consists of a
proportional controller and load elements (inertia and viscous-friction
elements)
We wish to control the output position in accordance with the input
position r.
The equation for the load elements is
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Where T is the torque produced by the proportional controller whose gain
is K.
Taking Laplace transforms of both sides of
this last equation, assuming the zero initial
conditions, we obtain
So the transfer function between and
is
The closed-loop transfer function is then obtained as
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Such a system where the closed-loop transfer function possesses
two poles is called a second-order system. (Some second-order
systems may involve one or two zeros.)
The closed-loop poles are complex conjugates if 4 0and
they are real if 4 0. In the transient-response analysis, it is
convenient to write
Where is called the attenuation; the undamped natural frequency; and
the damping ratio of the system. The damping ratio is the ratio of the
actual damping to the critical damping.
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The closed-loop transfer function / given by
This form is called the standard form of the second-order system.
The dynamic behavior of the second-order system can then be described in terms of
two parameters and . If 0 1, the closed-loop poles are complex
conjugates and lie in the left-half s plane. The system is then called underdamped,
and the transient response is oscillatory. If 0, the transient response does not die
out. If 1, the system is called critically damped. Overdamped systems
correspond to 1.
We shall now solve for the response of the system shown in Figure above to a unit-
step input.We shall consider three different cases: the underdamped (0 1),
critically damped ( 1), and overdamped 1 cases.
(1) Underdamped case 0 1): In this case, / can be
written
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Where 1 .The frequency is called the damped natural
frequency. For a unit-step input, C(s) can be written
Can be obtained easily if is written in the following form:
The Laplace transform
Hence the inverse Laplace transform is obtained as for
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it can be seen that the frequency of transient oscillation is the
damped natural frequency and thus varies with the damping ratio .The
error signal for this system is the difference between the input and output
and is
This error signal exhibits a damped sinusoidal oscillation. At steady state,
or at ∞, no error exists between the input and output.
The response ) for the zero damping case may be obtained by
substituting 0
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An increase in would reduce the damped natural frequency If is
increased beyond unity, the
response becomes overdamped and will not oscillate.
(2) Critically damped case : If the two poles of are equal,
the system is said to be a critically damped one. For a unit-step input,
and ) can be written
The inverse Laplace transform
(3) Overdamped case : In this case, the two poles of /
are negative real and unequal. For a unit-step input, 1/ and
can be written
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The inverse Laplace transform
Where 1 and 1 . Thus, the
response includes two decaying exponential terms.
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Note that two second-order systems having the same but different will exhibit the same overshoot
and the same oscillatory pattern. Such systems are said to have the same relative stability.
Definitions of Transient-Response Specifications.
Frequently, the performance characteristics of a control system are specified in terms
of the transient response to a unit-step input, since it is easy to generate and is
sufficiently drastic. (If the response to a step input is known, it is mathematically
possible to compute the response to any input.) The transient response of a system
to a unit-step input depends on the initial conditions. For convenience in comparing
transient responses of various systems, it is a common practice to use the standard
initial condition that the system is at rest initially with the output and all time
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derivatives thereof zero. Then the response characteristics of many systems can be
easily compared. The transient response of a practical control system often exhibits
damped oscillations before reaching steady state. In specifying the transient-
response characteristics of a control system to a unit-step input, it is common to
specify the following:
1. Delay time, :The delay time is the time required for the response to reach
half the final value the very first time.
2. Rise time, :The rise time is the time required for the response to rise from
10% to 90%, 5% to 95%, or 0% to 100% of its final value. For underdamped
second order systems, the 0%to 100%rise time is normally used. For
overdamped systems, the 10% to 90% rise time is commonly used.
3. Peak time, : The peak time is the time required for the response to reach
the first peak of the overshoot.
4. Maximum (percent) overshoot, Mp: The maximum overshoot is the
maximum peak value of the response curve measured from unity. If the final
steady-state value of the response differs from unity, then it is common to
use the maximum percent overshoot. It is defined by
The amount of the maximum (percent) overshoot directly indicates the relative
stability of the system
5. Settling time, : The settling time is the time required for the response
curve to reach and stay within a range about the final value of size specified
by absolute percentage of the final value (usually 2% or 5%). The settling
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time is related to the largest time constant of the control system. Which
percentage error criterion to use may be determined from the objectives of
the system design in question
The time-domain specifications just given are quite important, since most
control systems are time-domain systems; that is, they must exhibit
acceptable time responses. (This means that, the control system must be
modified until the transient response is satisfactory.)
Second-Order Systems and Transient-Response Specifications. In the
following, we shall obtain the rise time, peak time, maximum overshoot,
and settling time of the second-order system given by Equation:
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These values will be obtained in terms of and .The system is assumed
to be underdamped.
Rise time : we obtain the rise time by letting 1.
Since
Where angle is defined in Figure below clearly, for a small value of ,
must be large.
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Peak time : we may obtain the peak time by differentiating with respect to
time and letting this derivative equal zero. Since
And the cosine terms in this last equation cancel each other, , evaluated at
, can be simplified to.
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This last equation yields the following equation:
Maximum overshoot : The maximum overshoot occurs at the peak time or at
. Assuming that the final value of the output is unity, is obtained as
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Settling time : For an underdamped second-order system, the transient response
is obtained as
For convenience in comparing the responses of systems, we commonly define the
settling time to be
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Note that the settling time is inversely proportional to the product of the damping
ratio and the undamped natural frequency of the system. Since the value of is
usually determined from the requirement of permissible maximum overshoot, the
settling time is determined primarily by the undamped natural frequency . This
means that the duration of the transient period may be varied, without changing the
maximum overshoot, by adjusting the undamped natural frequency
From the preceding analysis, it is evident that for rapid response must be large.To
limit the maximum overshoot and to make the settling time small, the damping
ratio should not be too small. The relationship between the maximum percent
overshoot and the damping ratio is presented in Figure. Note that if the
damping ratio is between 0.4 and 0.7, then the maximum percent overshoot for step
response is between 25% and 4%.
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